Calculus Examples

$f (x) = \frac{4 t}{t + 5}$

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \frac{4 t}{t + 5}$

Step 2.1.2

The final answer is $\frac{4 t}{t + 5}$ .

$\frac{4 t}{t + 5}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{4 t}{t + 5}$

$f (x) = \frac{4 t}{t + 5}$

$f (x + h) = \frac{4 t}{t + 5}$

$f (x) = \frac{4 t}{t + 5}$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{\frac{4 t}{t + 5} - (\frac{4 t}{t + 5})}{h}$

Step 4

Simplify.

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Step 4.1

Subtract $\frac{4 t}{t + 5}$ from $\frac{4 t}{t + 5}$ .

$f' (x) = lim_{h \to 0} \frac{0}{h}$

Step 4.2

Divide $0$ by $h$ .

$f' (x) = lim_{h \to 0} 0$

Step 5

Evaluate the limit of $0$ which is constant as $h$ approaches $0$ .

$0$

Step 6